'Primeval slime' is a reference to the beginning of life - when cells formed, and started joining into multicellular organisms. 'Abyss' is a metaphorical suggestion of a hole - so when we evolved from single cells, we 'climbed up out of an abyss'. The suggestion is that when civilisation falls apart, humanity falls back down the evolutionary ladder - back where we came from. This is obviously an exaggeration, but that is the idea.

In this context, I would assume "the slime of the primeval abyss" means they are most likely decreasing to more early, primitive stages of life. It's like the opposite of progressing or evolution. Also, where it says "having climbed up from", that is referring too how they were once at that stage of life, however they progressed past that, and now they are returning to that previous stage.